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A is any language and $\mathrm{DROP}-\mathrm{OUT}\left(\mathrm{A}\right)=\left\{\mathrm{xz}\right|\mathrm{xyz}鈭�\mathrm{A}鈥�鈥�\mathrm{where}鈥�鈥�\mathrm{x},\mathrm{z}鈭�鈭�*,\mathrm{y}鈭�鈭�\}$.

We have to prove that the class of the regular languages closed under the $\mathrm{DROP}\_\mathrm{OUT}$ operation.

If A is a regular language, then $\mathrm{DROP}-\mathrm{OUT}\left(\mathrm{A}\right)$is also regular language.

We have to take that A is regular and we have to prove that the $\mathrm{DROP}-\mathrm{OUT}\left(\mathrm{A}\right)$is regular.

Since A is a regular language, it must be recognized by a DFA.

Let $M=\left(Q,鈭�,未,q_0,F\right)$be the DFA that recognizes A.

Now we will construct the NFA $N=\left(Q\text{'},鈭�鈭�\{鈭�\},未\text{'},q_0^{\text{'}},F\text{'}\right)$that recognizes $\mathrm{DROP}-\mathrm{OUT}\left(\mathrm{A}\right)$.

There are two copies of Machine M.

Copy 1:Copy 1 corresponds to the state of having 鈥榥ot yet skipped a symbol鈥�

Copy 2:Copy 2 corresponds to the state of having 鈥渁lready skipped a symbol鈥�.

(i) Proof by picture: -

$N=\left(Q\text{'},鈭�鈭�\{鈭�\},未\text{'},q_0^{\text{'}},F\text{'}\right)$

$Q=\left\{\left(q,b\right)\right|q鈭�Q,b鈭�\{0,1\}\}$= set of states

$q_0^{\text{'}}=$start state

$=\left({\mathrm{q}}_0,0\right)$

$\mathrm{F}\text{'}=$set of final states

$=\left\{\right(\mathrm{q},1|\mathrm{q}鈭�\mathrm{F}\}$.

$\text{未'}$is gives as follows:

localid="1663243041084" $鈫�鈥�鈥�未\text{'}\left(\right(a,b),a)=\left\{\right(未(q,a),b\left)\right\}鈭赌q鈭�Q,b鈭�\{0,1\},a鈭�鈭�$

This means that both the copy1 and copy2 of the machineMdo exactly as the original machine does on every symbol a of the alphabet

localid="1663243051646" $鈫�鈥�鈥�未\text{'}\left(\right(q,0),鈭�)=\left\{\right(\hat{q},1)鈭�a鈭�鈭�,未(q,a)=\hat{q}\}$

Also at every stage, the machine has the option to skip a character. The only accepting sates are in copy 2. This means, the machine cannot accept a string without skipping a character.

The formal proof is given by induction on the length of the string.

An appropriate inductive hypothesis is to assume that, for any string w of length k,

The machine M stays in the copy -1 if it has not yet skipped a symbol.

i.e. $未\text{'}*\left(\right(q_0,0),蝇)=(q_1,0)鈥�鈥�鈥�iff鈥�鈥�鈥�未(q_0,蝇)=q_1$

The machine M jumps to the copy-2 if there is some symbol a that is skipped.

i.e. $未\text{'}*\left(\right(q_0,0),蝇)=(q_1,1)鈥�鈥�鈥�if鈥�鈥�鈥�未(q_0,{蝇}_{1}a{蝇}_2)=q_1$ .

So, in both (i) and (ii) we constructed and NFAN that recognizes the language role="math" localid="1663242954164" $\mathrm{DROP}-\mathrm{OUT}\left(\mathrm{A}\right)$.

Thus $\mathrm{DROP}-\mathrm{OUT}\left(\mathrm{A}\right)$is regular.

Hence class of regular languages is closed under $\mathrm{DROP}-\mathrm{OUT}\left(\mathrm{A}\right)$operation.